Problem: Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{t^2 + 12t + 27}{t^2 - 4t - 21}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 + 12t + 27}{t^2 - 4t - 21} = \dfrac{(t + 9)(t + 3)}{(t - 7)(t + 3)} $ Notice that the term $(t + 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 3)$ gives: $z = \dfrac{t + 9}{t - 7}$ Since we divided by $(t + 3)$, $t \neq -3$. $z = \dfrac{t + 9}{t - 7}; \space t \neq -3$